The perceived usefulness of application servers seems unquestioned in many corners of the I.T. world. The conditions that facilitate the deployment of application servers in architectural planning persist. However some people think that code can be written to function on a server that interacts with a web and/or database server without a typical application server being deployed in the middle of the customized code and the OS itself. Some professionals do not think that application servers are necessary.
JBoss, Tomcat, TomEE, GlassFish and other application servers are common. They can require considerable operational staffing to support. There can be licensing costs (e.g., with IBM's WebSphere, Oracle's WebLogic). The effort of the developers can be considerable to ensure the customized program works with the application server. There can be vendor lock-in after the code is reliant upon the proprietary application server. The application server program can create additional overhead and resources that a finely-tuned program would not use. For low latency programming, an application server may not be an option.
It is noteworthy that the common LAMP stack involves an OS, a web server and front-end code, and a database server. There is no application server in this complementary group of technologies.
It is possible that as technology matures, that the market for application servers will solidify in certain enterprise needs. Some application servers will provide a niche benefit, and developers and/or enterprises will leverage certain features thereof.
If you want more information about why application servers may not be the wave of the future, see this article by Forrester. Another interesting article was written by this British subject matter expert.